The nattokinase scam

A conversation about vitamin K2 commonly leads to confusion. Several people have asked about something called nattokinase.

The scientific data on the potential role of vitamin K2 deficiency in causing both osteoporosis and vascular calcification is fascinating. Along with vitamin D3, vitamin K2 may be an important factor in regulation of calcium metabolism. Supplementation may prove to be a major strategy for inhibition of vascular calcification.

Obtaining K2 in the diet is tricky, since it's present in just a handful of foods: egg yolks, liver, traditional cheeses, and natto. This is where the confusion starts.

Natto is a Japanese fermented soy product. I've had it and it's quite disgusting. Nonetheless, Japanese who eat natto experience less fracture. (A parallel study in heart disease has not been performed.) Natto is also a source of another substance called nattokinase.

Advocates (othe! rwise often known as supplement distributors) claim that nattokinase is a "fibrinolytic", or blood clot-dissolving, preparation that "improves blood flow, protects from blood clots, and prevents heart attacks and strokes."

Don't you believe it. This is patent nonsense. There are several problems with this rationale:

--Any oral fibrinolytic agent is promptly degraded in the highly acid environment of the stomach. That's why all medically used fibrinolytics are given intravenously. Drug companies have struggled for years to encapsulate, modify, or somehow protect protein (or polypeptide) products taken orally from degrading this way. They've never succeeded. That's why, for instance, growth hormone (a polypeptide) remains an injection, not an oral agent. An oral growth hormone, by the way, would sell like mad, so the drug companies would very much like to figure out how to bypass the degradative effects of stomach acid. One of the "researchers" behind th! e nattokinase claims boasts that he has single-handedly figure! d out ho w to protect the nattokinase molecule in the gastrointestinal tract. However, he won't tell anybody how he does it. Right.

--Fibrinolytic agents are extremely dangerous. In years past, we used to treat heart attacks with intravenous fibrinolytic agents like tissue plasminogen activator, urokinase, streptokinase, and others. They have fallen by the wayside, for the most part, because of limited effectiveness and the unavoidable dangers of their use. Fibrinolytics are "dumb": they dissolve blood clots in both good places and bad. While they might dissolve the blood clot causing your heart attack, they also degrade the tiny clot in your cerebral (brain) circulation that was protective. That's why fatal brain hemorrhages, bleeding stomach ulcers, and blood oozing from strange places can also occur with fibrinolytic administration. Believe me, I've seen it happen, and I've watched people die from them.

The idea that a small dose taken orally is healthy is! ridiculous. Even if nattokinase worked, why the heck would you take an agent that has known dangerous and very real consequences?

Don't let this idiocy reflect poorly on the K2 conversation, which, I believe, holds real merit and is backed by legitimate science. This is symptomatic of a larger difficulty with the supplement industry: Insane and unfounded claims about one supplement erodes credibility for the entire industry. It gives regulation-crazed people like the FDA ammunition to go after supplements, something none of us need. You and I have to sift through the nonsense to uncover the real gems in this rockpile, real gems like vitamin D3, omega-3 fatty acids from fish oil, and, perhaps, vitamin K2. But not nattokinase.

scan valid interval

Probability and Statistics 101: Pre-lesson lesson

This isn't even the first installment in the curriculum. It's something I'm going to hand out to read before the first class. Many of you will find it completely unnecessary. Some may feel insulted by it, but please don't be: lots of people out there actually, truly need to read this. If you feel you don't, read it anyway as a lesson in what lots of people actually need to read. If you are matho-phobic or hate math or think you just can't possibly understand it, this will help. I hope.

In public health science, and for that matter in most fields of scientific inquiry, we frequently talk about proportions, rates or percentages.

These all mean exactly the same thing!

When people speak of proportions, there is a tendency to represent them as decimals, like this: .9 (pronounced "point nine.") When people speak of rates, they are probably a bit more likely to represent them as fractions, like this: 9/10 (pronounced "nine tenths," or "ni! ne out of ten.") To get the percentage, you just multiply by 100: 90% (pronounced "ninety percent.")

They all mean exactly the same thing! There is no difference at all. I would not be doing any violence to the English language or to mathematics to speak of a proportion of .9, a proportion of 9/10, or a proportion of 90%. There is no difference at all, not even a little tiny difference. None.

For some reason, when I show people a percentage -- 90%, 87%, whatever -- they usually say that's fine, they get it. But when I show them .9, or .87, the very same people often say they don't get it, they can't possibly understand it, it's math and it's just too confusing. That is like saying you understand me perfectly well when I say a Ford is a car, but it's impossible to understand or follow what I'm saying if I say a Ford is an automobile.

There's nothing to understand. There's nothing to "get." There is no deep meaning or special attri! bute of the symbol .9 that makes it any different from the sym! bol 90%. If you are at all confused, just keep in mind that the space just to the right of the decimal point is 10ths. .9 = 9 tenths = 9/10 = 90/100 = 90%. The next space is hundredths. .97 = 9 tenths + 7 100ths = 90 100ths + 7 100ths = 97 100ths = 97%. That's all there is to it. If you still think you don't get it, it's because you are convinced there must be more to it than that, that there must be something I'm not telling you, some deeper secret. There isn't.

Probabilities: Probabilities are a lot like proportions. If 90% or .9 or 9/10ths of the people in the room are right handed, and I pick somebody at random, the probability that person will be right handed is 90%, .9, or 9/10. We will often say "nine out of ten" for a probability but it means exactly the same thing to say .9 or 9/10 or 90%. It obviously saves space to write .9 and it makes calculations easier, so that's what I will do most of the time. If this b! others you, you can think "ninety percent" every time you see .9, and it will be just fine.

Formulas and equations: Mathematical formulas and equations give lots of people the heebie jeebies. As soon as they see one, they stop reading and they stop thinking. One reason for this, I think, is the convention, when writing about mathematical subjects, of first writing down the equation, and then telling the reader what the symbols mean. The result is that you'll see a bit fat equation sitting in the middle of the page, and you have no idea what it means, so you feel stupid. But remember, at this point, even Einstein has no idea what it means. You have to read the next sentence, where the person tells you what it means, and look up at the equation and back and forth between the equation and the sentence underneath it to get what all the symbols mean. For example, suppose I write this:

P(RH) = .9

You're baffled and you're mad at me! because you don't know what that means. But all you had to do was wait for me to tell you. "P" means the probability of whatever term is enclosed in the parentheses right after it. "RH" means being right-handed. So P(RH)=.9 means "The probability of being right-handed is point nine" (or ninety percent if you prefer).

Why didn't I just say so in the first place? Because I want to use this idea more generally: I want to use this structure in arguments and equations where I'm not just talking about being right-handed and .9, but whatever phenomenon and whatever probability happens to apply. The equation is a handy structure that doesn't take a lot of space to write down and captures the ideas I want to talk about, not just the specific case.

So what I am going to do is first, discuss concepts qualitatively, in prose, with as little use of mathematical symbols as I can get away with. I'll only use mathematical symbo! ls when I'm pretty sure it will make it easier, but we're going to be talking about mathematics so it will be impossible to avoid referring to mathematical concepts such as one thing being equal to another.

Then I'll use one or more specific examples and maybe do some arithmetic with them to show you how the idea works.

Then, and only then, I will present general formulas that embody the idea. If you don't like that part, you can skip it. It doesn't need to turn you off to the rest of it.

Any questions? Are we ready to begin?

probability equations

About AIEEE Exam

INTRODUCTION:
Programme of Action (PoA), 1992 under the National Policy on Education (NPE), 1986 envisaged conduct of a common entrance examination on all India basis for admission to professional and technical programmes in the country. For admission to Engineering and Architecture/Planning programmes, Government of India vide Resolution dated 18th October 2001 has laid down a Three – Exam Scheme (JEE and AIEEE at the National Level and the State Level Engineering Entrance Examinations (SLEEE) for State Level Institutions – with an option to join AIEEE). This takes care of varying admission standards in these programmes and helps in maintenance of professional standards. This also solves problems of overlaps and reduces physical, mental and financial burden on students and their parents due to multiplicity of entrance examinations.
The eighth All India Engineering Entrance Examination (AIEEE) was held on 26th April' 2009. 9,62,119 candidates appeared, out of 10,10,061 candidates registered, for the examination at 1,460 centres located in 85 cities. Approximately, 23,676 seats for B E/B.Tech and 855 seats for B.Arch/ B. Planning in various institutions, namely National Institutes of Technology, Deemed Universities, Technical Institutions and other Govt. funded Institutions were offered through this examination. A centralized counselling had been organized for filling up the All India seats.
The States/Institutes listed below used AIEEE-2009 ranks to fill seats through their own counselling:- West Bengal, Himachal Pradesh, Haryana, Uttarakhand, Punjab University, Chandigarh, Punjab Engineering College, Chandigarh, Dhirubhai Ambani Institute of Information and Communication Technology, Gandhinagar, Army Institute of Technology, Pune, 15% all India quota for D C E/ N.S.I.T, Delhi under Delhi University.
Keeping in view greater variety of subjects offered by different School Boards at the 10+2 (Class – XII) level, more subject combinations in qualifying examination (10+2) would now be possible. For the convenience of candidates, provision for on-line submission of Application Form has also been made.



Application Procedure:
Application can either be made  ‘Online’ or submitted on prescribed Application Form only.Information Bulletin containing Application Form can also be obtained by post from the Assistant Secretary (AIEEE), Central Board of Secondary Education, PS 1-2, Institutional Area, I P Extension, Patparganj, Delhi - 110092 or personally from Regional Offices of CBSE and Designated Branches of Syndicate Bank/other Banks/Designated Institutions (List given in Appendix IV). The Syndicate Bank/other Banks, the designated institutions and Regional Offices of the CBSE will not send the Information Bulletin by Post. Hence, no request should be made to them for sending the Bulletin by Post. Candidate should not buy the Information Bulletin from any private publisher/ vendor as these may be fake Information Bulletins.
The Regional Offices of CBSE, designated branches of Syndicate Banks/Other Banks/designated Institutions (see list given in Appendix-IV) are not responsible for any consequences, that may arise due to non-availability of forms in case the copies of application forms are sold out before the last date. The candidate should get the Information Bulletin containing application forms from the Assistant Secretary (AIEEE), CBSE, PS 1-2 Institutional Area, IP Extension, Patparganj, Delhi-110 092 by post. The cost of Information Bulletin inclusive of examination and counselling fee for BE/B.Tech only or B.Arch/ B.Planning only is Rs. 450/- for General and OBC candidates and Rs. 225/- for SC/ST candidates. Candidates appearing for both BE/B.Tech and B.Arch/B.Planning together should send their application form along with additional fee in the form of Demand Draft of Rs. 300/- for General & OBC candidates and Rs. 150/- for SC/ST candidates in favour of Secretary, CBSE payable at Del! hi/ New Delhi.
For candidates opting examination centre in foreign country, the examination and counselling fees for appearing in BE/B Tech only or B Arch/B Planning only is Rs. 1000/- for General and OBC candidates and Rs. 500/- for SC/ST candidates that includes counselling fee. They should send additional fees @ Rs. 550/- for General and OBC Category & Rs. 275/- for SC & ST Category in the form of demand draft in favour of Secretary, C.B.S.E payable at Delhi/New Delhi. Those who want to appear both BE/B.Tech & B. Arch/B Planning together they should send their application form along with additional fee in the form of demand Draft of Rs.1550/- for General and OBC candidates and Rs. 775/ - for SC/ST candidates in favour of Secretary, CBSE payable at Delhi/New Delhi.


Syllabus:

MATHEMATICS
UNIT 1: SETS, RELATIONS AND FUNCTIONS:
Sets and their representation; Union, intersection and complement of sets and their algebraic properties; Power set; Relation, Types of relations, equivalence relations, functions;. one-one, into and onto functions, composition of functions.

UNIT 2: COMPLEX NUMBERS AND QUADRATIC EQUATIONS:
Complex numbers as ordered pairs of reals, Representation of complex numbers in the form a+ib and their representation in a plane, Argand diagram, algebra of complex numbers, modulus and argument (or amplitude) of a complex number, square root of a complex number, triangle inequality, Quadratic equations in real and complex number system and their solutions. Relation between roots and co-efficients, nature of roots, formation of quadratic equations with given roots.
UNIT 3: MATRICES AND DETERMINANTS:
Matrices, algebra of matrices, types of matrices, determinants and matrices of order two and three. Properties of determinants, evaluation of determinants, area of triangles using determinants. Adjoint and evaluation of inverse of a square matrix using determinants and elementary transformations, Test of consistency and solution of simultaneous linear equations in two or three variables using determinants and matrices.

UNIT 4: PERMUTATIONS AND COMBINATIONS:
Fundamental principle of counting, permutation as an arrangement and combination as selection, Meaning of P (n,r) and C (n,r), simple applications.

UNIT 5: MATHEMATICAL INDUCTION:
Principle of Mathematical Induction and its simple applications.
UNIT 6: BINOMIAL THEOREM AND ITS SIMPLE APPLICATIONS:
Binomial theorem for a positive integral index, general term and middle term, properties of Binomial coefficients and simple applications.

UNIT 7: SEQUENCES AND SERIES:
Arithmetic and Geometric progressions, insertion of arithmetic, geometric means between two given numbers. Relation between A.M. and G.M. Sum upto n terms of special series: Sn, Sn2, Sn3. Arithmetico - Geometric progression.

UNIT 8: LIMIT, CONTINUITY AND DIFFERENTIABILITY:
Real - valued functions, algebra of functions, polynomials, rational, trigonometric, logarithmic and exponential functions, inverse functions. Graphs of simple functions. Limits, continuity and differentiability. Differentiation of the sum, difference, product and quotient of two functions. Differentiation of trigonometric, inverse trigonometric, logarithmic, exponential, composite and implicit functions; derivatives of order upto two. Rolle’s and Lagrange’s Mean Value Theorems. Applications of derivatives: Rate of change of quantities, monotonic - increasing and decreasing functions, Maxima and minima of functions of one variable, tangents and normals.

UNIT 9: INTEGRAL CALCULUS:
Integral as an anti - derivative. Fundamental integrals involving algebraic, trigonometric, exponential and logarithmic functions. Integration by substitution, by parts and by partial fractions. Integration using trigonometric identities.
Evaluation of simple integrals of the type



Integral as limit of a sum. Fundamental Theorem of Calculus. Properties of definite integrals. Evaluation of definite integrals, determining areas of the regions bounded by simple curves in standard form.
UNIT 10: DIFFERENTIAL EQUATIONS:
Ordinary differential equations, their order and degree. Formation of differential equations. Solution of differential equations by the method of separation of variables, solution of homogeneous and linear differential equations of the type:
dy
-- + p (x) y = q (x)
dx
UNIT 11: CO-ORDINATE GEOMETRY:
Cartesian system of rectangular co-ordinates in a plane, distance formula, section formula, locus and its equation, translation of axes, slope of a line, parallel and perpendicular lines, intercepts of a line on the coordinate axes.
Straight lines
Various forms of equations of a line, intersection of lines, angles between two lines, conditions for concurrence of three lines, distance of a point from a line, equations of internal and external bisectors of angles between two lines, coordinates of centroid, orthocentre and circumcentre of a triangle, equation of family of lines passing through the point of intersection of two lines.
Circles, conic sections
Standard form of equation of a circle, general form of the equation of a circle, its radius and centre, equation of a circle when the end points of a diameter are given, points of intersection of a line and a circle with the centre at the origin and condition for a line to be tangent to a circle, equation of the tangent. Sections of cones, equations of conic sections (parabola, ellipse and hyperbola) in standard forms, condition for y = mx + c to be a tangent and point (s) of tangency.

UNIT 12: THREE DIMENSIONAL GEOMETRY:
Coordinates of a point in space, distance between two points, section formula, direction ratios and direction cosines, angle between two intersecting lines. Skew lines, the shortest distance between them and its equation. Equations of a line and a plane in different forms, intersection of a line and a plane, coplanar lines.

UNIT 13: VECTOR ALGEBRA:
Vectors and scalars, addition of vectors, components of a vector in two dimensions and three dimensional space, scalar and vector products, scalar and vector triple product.

UNIT 14:
STATISTICS AND PROBABILITY:
Measures of Dispersion: Calculation of mean, median, mode of grouped and ungrouped data. Calculation of standard deviation, variance and mean deviation for grouped and ungrouped data.
Probability: Probability of an event, addition and multiplication theorems of probability, Baye’s theorem, probability distribution of a random variate, Bernoulli trials and Binomial distribution.

UNIT 15: TRIGONOMETRY:
Trigonometrical identities and equations. Trigonometrical functions. Inverse trigonometrical functions and their properties. Heights and Distances.

UNIT 16: MATHEMATICAL REASONING:
Statements, logical operations and, or, implies, implied by, if and only if. Understanding of tautology, contradiction, converse and contrapositive.

PHYSICS
      The syllabus contains two Sections - A and B. Section - A pertains to the Theory Part having 80% weightage, while Section - B contains Practical Component (Experimental Skills) having 20% weightage.
SECTION A

UNIT 1: PHYSICS AND MEASUREMENT
Physics, technology and society, S I units, Fundamental and derived units. Least count, accuracy and precision of measuring instruments, Errors in measurement, Significant figures. Dimensions of Physical quantities, dimensional analysis and its applications.

UNIT 2: KINEMATICS
Frame of reference. Motion in a straight line: Position-time graph, speed and velocity. Uniform and non-uniform motion, average speed and instantaneous velocity Uniformly accelerated motion, velocity-time, position- time graphs, relations for uniformly accelerated motion. Scalars and Vectors, Vector addition and Subtraction, Zero Vector, Scalar and Vector products, Unit Vector, Resolution of a Vector. Relative Velocity, Motion in a plane, Projectile Motion, Uniform Circular Motion.

UNIT 3: LAWS OF MOTION
Force and Inertia, Newton’s First Law of motion; Momentum, Newton’s Second Law of motion; Impulse; Newton’s Third Law of motion. Law of conservation of linear momentum and its applications, Equilibrium of concurrent forces.
Static and Kinetic friction, laws of friction, rolling friction.
Dynamics of uniform circular motion: Centripetal force and its applications.

UNIT 4: WORK, ENERGY AND POWER
Work done by a constant force and a variable force; kinetic and potential energies, workenergy theorem, power. Potential energy of a spring, conservation of mechanical energy, conservative and nonconservative forces; Elastic and inelastic collisions in one and two dimensions.

UNIT 5: ROTATIONAL MOTION
Centre of mass of a two-particle system, Centre of mass of a rigid body; Basic concepts of rotational motion; moment of a force, torque, angular momentum, conservation of angular momentum and its applications; moment of inertia, radius of gyration. Values of moments of inertia for simple geometrical objects, parallel and perpendicular axes theorems and their applications. Rigid body rotation, equations of rotational motion.

UNIT 6: GRAVITATION
The universal law of gravitation. Acceleration due to gravity and its variation with altitude and depth. Kepler’s laws of planetary motion. Gravitational potential energy; gravitational potential. Escape velocity. Orbital velocity of a satellite. Geo-stationary satellites.

UNIT 7: PROPERTIES OF SOLIDS AND LIQUIDS
Elastic behaviour, Stress-strain relationship, Hooke’s Law, Young’s modulus, bulk modulus, modulus of rigidity. Pressure due to a fluid column; Pascal’s law and its applications. Viscosity, Stokes’ law, terminal velocity, streamline and turbulent flow, Reynolds number. Bernoulli’s principle and its applications. Surface energy and surface tension, angle of contact, application of surface tension - drops, bubbles and capillary rise. Heat, temperature, thermal expansion; specific heat capacity, calorimetry; change of state, latent heat. Heat transfer- conduction, convection and radiation, Newton’s law of cooling.
UNIT 8: THERMODYNAMICS
Thermal equilibrium, zeroth law of thermodynamics, concept of temperature. Heat, work and internal energy. First law of thermodynamics. Second law of thermodynamics: reversible and irreversible processes. Carnot engine and its efficiency.
UNIT 9: KINETIC THEORY OF GASES
Equation of state of a perfect gas, work doneon compressing a gas.Kinetic theory of gases - assumptions, concept of pressure. Kinetic energy and temperature: rms speed of gas molecules; Degrees of freedom, Law of equipartition of energy,applications to specific heat capacities of gases; Mean free path, Avogadro’s number.
UNIT 10: OSCILLATIONS AND WAVES
Periodic motion - period, frequency, displacement as a function of time. Periodic functions. Simple harmonic motion (S.H.M.) and its equation; phase; oscillations of a spring -restoring force and force constant; energy in S.H.M. - kinetic and potential energies; Simple pendulum - derivation of expression for its time period; Free, forced and damped oscillations, resonance.

Wave motion. Longitudinal and transverse waves, speed of a wave. Displacement relation for a progressive wave. Principle of superposition of waves, reflection of waves, Standing waves in strings and organ pipes, fundamental mode and harmonics, Beats, Doppler effect in sound

UNIT 11: ELECTROSTATICS
Electric charges: Conservation of charge, Coulomb’s law-forces between two point charges, forces between multiple charges; superposition principle and continuous charge distribution.
Electric field: Electric field due to a point charge, Electric field lines, Electric dipole, Electric field due to a dipole, Torque on a dipole in a uniform electric field.

Electric flux, Gauss’s law and its applications to find field due to infinitely long uniformly charged straight wire, uniformly charged infinite plane sheet and uniformly charged thin spherical shell. Electric potential and its calculation for a point charge, electric dipole and system of charges; Equipotential surfaces, Electrical potential energy of a system of two point charges in an electrostatic field.

Conductors and insulators, Dielectrics and electric polarization, capacitor, combination of capacitors in series and in parallel, capacitance of a parallel plate capacitor with and without dielectric medium between the plates, Energy stored in a capacitor.

UNIT  12: CURRENT ELECTRICITY
Electric current, Drift velocity, Ohm’s law, Electrical resistance, Resistances of different materials, V-I characteristics of Ohmic and nonohmic conductors, Electrical energy and power, Electrical resistivity, Colour code for resistors; Series and parallel combinations of resistors; Temperature dependence of resistance.

Electric Cell and its Internal resistance, potential difference and emf of a cell, combination of cells in series and in parallel. Kirchhoff’s laws and their applications. Wheatstone bridge, Metre bridge. Potentiometer - principle and its applications.

UNIT  13: MAGNETIC EFFECTS OF CURRENT AND MAGNETISM
Biot - Savart law and its application to current carrying circular loop. Ampere’s law and its applications to infinitely long current carrying straight wire and solenoid. Force on a moving charge in uniform magnetic and electric fields. Cyclotron.

Force on a current-carrying conductor in a uniform magnetic field. Force between two parallel current-carrying conductors-definition of ampere. Torque experienced by a current loop in uniform magnetic field; Moving coil galvanometer, its current sensitivity and conversion to ammeter and voltmeter.

Current loop as a magnetic dipole and its magnetic dipole moment. Bar magnet as an equivalent solenoid, magnetic field lines; Earth’s magnetic field and magnetic elements. Para-, dia- and ferro- magnetic substances.

Magnetic susceptibility and permeability, Hysteresis, Electromagnets and permanent magnets.

UNIT 14: ELECTROMAGNETIC INDUCTION AND ALTERNATING CURRENTS
Electromagnetic induction; Faraday’s law, induced emf and current; Lenz’s Law, Eddy currents. Self and mutual inductance. Alternating currents, peak and rms value of alternating current/ voltage; reactance and impedance; LCR series circuit, resonance; Quality factor, power in AC circuits, wattless current. AC generator and transformer.

UNIT 15: ELECTROMAGNETIC WAVES
Electromagnetic waves and their characteristics. Transverse nature of electromagnetic waves.

Electromagnetic spectrum (radio waves, microwaves, infrared, visible, ultraviolet, Xrays, gamma rays). Applications of e.m. waves.

UNIT 16: OPTICS
Reflection and refraction of light at plane and spherical surfaces, mirror formula, Total internal reflection and its applications, Deviation and Dispersion of light by a prism, Lens Formula, Magnification, Power of a Lens, Combination of thin lenses in contact, Microscope and Astronomical Telescope (reflecting and refracting) and their magnifyingpowers.

Wave optics: wavefront and Huygens’ principle, Laws of reflection and refraction using Huygen’s principle. Interference, Young’s double slit experiment and expression for fringe width, coherent sources and sustained interference of light. Diffraction due to a single slit, width of central maximum. Resolving power of microscopes and astronomical telescopes, Polarisation, plane polarized light; Brewster’s law, uses of plane polarized light and Polaroids.

UNIT 17: DUAL NATURE OF MATTER AND RADIATION
Dual nature of radiation. Photoelectric effect, Hertz and Lenard’s observations; Einstein’s photoelectric equation; particle nature of light. Matter waves-wave nature of particle, de Broglie relation. Davisson-Germer experiment.

UNIT 18: ATOMS AND NUCLEI
Alpha-particle scattering experiment; Rutherford’s model of atom; Bohr model, energy levels, hydrogen spectrum. Composition and size of nucleus, atomic masses, isotopes, isobars; isotones. Radioactivity-alpha, beta and gamma particles/rays and their properties; radioactive decay law. Mass-energy relation, mass defect; binding energy per nucleon and its variation with mass number, nuclear fission and fusion.

UNIT 19: ELECTRONIC DEVICES
Semiconductors; semiconductor diode: I-V characteristics in forward and reverse bias; diode as a rectifier; I-V characteristics of LED, photodiode, solar cell and Zener diode; Zener diode as a voltage regulator. Junction transistor, transistor action, characteristics of a transistor; transistor as an amplifier (common emitter configuration) and oscillator. Logic gates (OR, AND, NOT, NAND and NOR). Transistor as a switch.

 
UNIT 20: COMMUNICATION SYSTEMS
  Propagation of electromagnetic waves in the atmosphere; Sky and space wave propagation, Need for modulation, Amplitude and Frequency Modulation, Bandwidth of signals, Bandwidth of Transmission medium, Basic Elements of a Communication System (Block Diagram only)
 
 
SECTION-B

UNIT 21: EXPERIMENTAL SKILLS
  Familiarity with the basic approach and observations of the experiments and activities:
1. Vernier callipers-its use to measure internal and external diameter and depth of a vessel.
2. Screw gauge-its use to determine thickness/diameter of thin sheet/wire.
3. Simple Pendulum-dissipation of energy by plotting a graph between square of amplitude and time.
4. Metre Scale - mass of a given object by principle of moments.
5. Young’s modulus of elasticity of the material of a metallic wire.
6. Surface tension of water by capillary rise and effect of detergents.
7. Co-efficient of Viscosity of a given viscous liquid by measuring terminal velocity of a given spherical body.
8. Plotting a cooling curve for the relationship between the temperature of a hot body and time.
9. Speed of sound in air at room temperature using a resonance tube. 10. Specific heat capacity of a given
(i) solid and
(ii) liquid by method of mixtures.
11. Resistivity of the material of a given wire using metre bridge.
12. Resistance of a given wire using Ohm’s law.
13. Potentiometer –
(i) Comparison of emf of two primary cells.
(ii) Determination of internal resistance of a cell.
14. Resistance and figure of merit of a galvanometer by half deflection method.
15. Focal length of:
(i) Convex mirror
(ii) Concave mirror, and
(iii) Convex lens using parallax method.
16. Plot of angle of deviation vs angle of incidence for a triangular prism.
17. Refractive index of a glass slab using a travelling microscope.
18. Characteristic curves of a p-n junction diode in forward and reverse bias.
19. Characteristic curves of a Zener diode and finding reverse break down voltage.
20. Characteristic curves of a transistor and finding current gain and voltage gain.
21. Identification of Diode, LED, Transistor, IC, Resistor, Capacitor from mixed collection of such items.
22. Using multimeter to:
(i) Identify base of a transistor
(ii) Distinguish between npn and pnp type transistor
(iii) See the unidirectional flow of current in case of a diode and an LED.
(iv) Check the correctness or otherwise of a given electronic component (diode, transistor or IC).
 
CHEMISTRY
 
SECTION-A

 
PHYSICAL CHEMISTRY

UNIT 1: SOME BASIC CONCEPTS IN CHEMISTRY
Matter and its nature, Dalton’s atomic theory; Concept of atom, molecule, element and compound; Physical quantities and their measurements in Chemistry, precision and accuracy, significant figures, S.I. Units, dimensional analysis; Laws of chemical combination; Atomic and molecular masses, mole concept, molar mass, percentage composition, empirical and molecular formulae; Chemical equations and stoichiometry.

UNIT 2: STATES OF MATTER
Classification of matter into solid, liquid and gaseous states.
Gaseous State:
Measurable properties of gases; Gas laws - Boyle’s law, Charle’s law, Graham’s law of diffusion, Avogadro’s law, Dalton’s law of partial pressure; Concept of Absolute scale of temperature; Ideal gas equation, Kinetic theory of gases (only postulates); Concept of average, root mean square and most probable velocities; Real gases, deviation from Ideal behaviour, compressibility factor, van der Waals equation, liquefaction of gases, critical constants.
Liquid State:
Properties of liquids - vapour pressure, viscosity and surface tension and effect of temperature on them (qualitative treatment only).
Solid State:
Classification of solids: molecular, ionic, covalent and metallic solids, amorphous and crystalline solids (elementary idea); Bragg’s Law and its applications; Unit cell and lattices, packing in solids (fcc, bcc and hcp lattices), voids, calculations involving unit cell parameters, imperfection in solids; Electrical, magnetic and dielectric properties.

UNIT 3: ATOMIC STRUCTURE
Discovery of sub-atomic particles (electron, proton and neutron); Thomson and Rutherford atomic models and their limitations; Nature of electromagnetic radiation, photoelectric effect; Spectrum of hydrogen atom, Bohr model of hydrogen atom - its postulates, derivation of the relations for energy of the electron and radii of the different orbits, limitations of Bohr’s model; Dual nature of matter, de-Broglie’s relationship, Heisenberg uncertainty principle. Elementary ideas of quantum mechanics, quantum mechanical model of atom, its important features, * and *2, concept of atomic orbitals as one electron wave functions; Variation of  *  and * 2 with r for 1s and 2s orbitals; various quantum numbers (principal, angular momentum and magnetic quantum numbers) and their significance; shapes of s, p and d - orbitals, electron spin and spin quantum number; Rules for filling electrons in orbitals – aufbau principle, Pauli’s exclusion principle and H! und’s rule, electronic configuration of elements, extra stability of half-filled and completely filled orbitals.

UNIT 4: CHEMICAL BONDING AND MOLECULAR STRUCTURE
Kossel - Lewis approach to chemical bond formation, concept of ionic and covalent bonds.
Ionic Bonding: Formation of ionic bonds, factors affecting the formation of ionic bonds; calculation of lattice enthalpy.
Covalent Bonding: Concept of electronegativity, Fajan’s rule, dipole moment; Valence Shell Electron Pair Repulsion (VSEPR) theory and shapes of simple molecules.
Quantum mechanical approach to covalent bonding: Valence bond theory - Its important features, concept of hybridization involving s, p and d orbitals; Resonance.
Molecular Orbital Theory - Its important features, LCAOs, types of molecular orbitals (bonding, antibonding), sigma and pi-bonds, molecular orbital electronic configurations of homonuclear diatomic molecules, concept of bond order, bond length and bond energy.
Elementary idea of metallic bonding. Hydrogen bonding and its applications.
 
UNIT 5: CHEMICAL THERMODYNAMICS
Fundamentals of thermodynamics: System and surroundings, extensive and intensive properties, state functions, types of processes.
First law of thermodynamics - Concept of work, heat internal energy and enthalpy, heat capacity, molar heat capacity; Hess’s law of constant heat summation; Enthalpies of bond dissociation, combustion, formation, atomization, sublimation, phase transition, hydration, ionization and solution.
Second law of thermodynamics- Spontaneity of processes; DS of the universe and DG of the system as criteria for spontaneity, DGo (Standard Gibbs energy change) and equilibrium constant.
UNIT 6: SOLUTIONS
Different methods for expressing concentration of solution - molality, molarity, mole fraction, percentage (by volume and mass both), vapour pressure of solutions and Raoult’s Law - Ideal and non-ideal solutions, vapour pressure - composition, plots for ideal and non-ideal solutions; Colligative properties of dilute solutions - relative lowering of vapour pressure, depression of freezing point, elevation of boiling point and osmotic pressure; Determination of molecular mass using colligative properties; Abnormal value of molar mass, van’t Hoff factor and its significance.
UNIT 7: EQUILIBRIUM
Meaning of equilibrium, concept of dynamic equilibrium.
Equilibria involving physical processes: Solid -liquid, liquid - gas and solid - gas equilibria, Henry’s law, general characterics of equilibrium involving physical processes.
Equilibria involving chemical processes: Law of chemical equilibrium, equilibrium constants (Kp and Kc) and their significance, significance of DG and DGo in chemical equilibria, factors affecting equilibrium concentration, pressure, temperature, effect of catalyst; Le Chatelier’s principle.
Ionic equilibrium: Weak and strong electrolytes, ionization of electrolytes, various concepts of acids and bases (Arrhenius, Br?nsted - Lowry and Lewis) and their ionization, acid - base equilibria (including multistage ionization) and ionization constants, ionization of water, pH scale, common ion effect, hydrolysis of salts and pH of their solutions, solubility of sparingly soluble salts and solubility products, buffer solutions.
UNIT   8: REDOX REACTIONS AND ELECTROCHEMISTRY
Electronic concepts of oxidation and reduction, redox reactions, oxidation number, rules for assigning oxidation number, balancing of redox reactions.
Eectrolytic and metallic conduction, conductance in electrolytic solutions, specific and molar conductivities and their variation with concentration: Kohlrausch’s law and its applications.
Electrochemical cells - Electrolytic and Galvanic cells, different types of electrodes, electrode potentials including standard electrode potential, half - cell and cell reactions, emf of a Galvanic cell and its measurement; Nernst equation and its applications; Relationship between cell potential and Gibbs’ energy change; Dry cell and lead accumulator; Fuel cells; Corrosion and its prevention.
UNIT 9: CHEMICAL KINETICS
Rate of a chemical reaction, factors affecting the rate of reactions: concentration, temperature, pressure and catalyst; elementary and complex reactions, order and molecularity of reactions, rate law, rate constant and its units, differential and integral forms of zero and first order reactions, their characteristics and half - lives, effect of temperature on rate of reactions - Arrhenius theory, activation energy and its calculation, collision theory of bimolecular gaseous reactions (no derivation).
UNIT 10:
SURFACE CHEMISTRY
Adsorption- Physisorption and chemisorption and their characteristics, factors affecting adsorption of gases on solids - Freundlich and Langmuir adsorption isotherms, adsorption from solutions.
Catalysis - Homogeneous and heterogeneous, activity and selectivity of solid catalysts, enzyme catalysis and its mechanism.
Colloidal state - distinction among true solutions, colloids and suspensions, classification of colloids - lyophilic, lyophobic; multi molecular, macromolecular and associated colloids (micelles), preparation and properties of colloids - Tyndall effect, Brownian movement, electrophoresis, dialysis, coagulation and flocculation; Emulsions and their characteristics.

SECTION-B

INORGANIC CHEMISTRY



UNIT 11: CLASSIFICATON OF ELEMENTS AND PERIODICITY IN PROPERTIES
Modem periodic law and present form of the periodic table, s, p, d and f block elements, periodic trends in properties of elements atomic and ionic radii, ionization enthalpy, electron gain enthalpy, valence, oxidation states and chemical reactivity.
UNIT 12: GENERAL PRINCIPLES AND PROCESSES OF ISOLATION OF METALS
Modes of occurrence of elements in nature, minerals, ores; steps involved in the extraction of metals - concentration, reduction (chemical. and electrolytic methods) and refining with special reference to the extraction of Al, Cu, Zn and Fe; Thermodynamic and electrochemical principles involved in the extraction of metals.
UNIT 13: HYDROGEN
Position of hydrogen in periodic table, isotopes, preparation, properties and uses of hydrogen; Physical and chemical properties of water and heavy water; Structure, preparation, reactions and uses of hydrogen peroxide; Classification of hydrides - ionic, covalent and interstitial; Hydrogen as a fuel.
UNIT 14: S - BLOCK ELEMENTS (ALKALI AND ALKALINE EARTH METALS)
Group - 1 and 2 Elements
General introduction, electronic configuration and general trends in physical and chemical properties of elements, anomalous properties of the first element of each group, diagonal relationships.
Preparation and properties of some important compounds - sodium carbonate, sodium chloride, sodium hydroxide and sodium hydrogen carbonate; Industrial uses of lime, limestone, Plaster of Paris and cement; Biological significance of Na, K, Mg and Ca.
UNIT 15: P - BLOCK ELEMENTS
Group - 13 to Group 18 Elements
General Introduction: Electronic configuration and general trends in physical and chemical properties of elements across the periods and down the groups; unique behaviour of the first element in each group.
Groupwise study of the p – block elements Group - 13
                Preparation, properties and uses of boron and aluminium; Structure, properties and uses of borax, boric acid, diborane, boron trifluoride, aluminium chloride and alums.
Group - 14
                Tendency for catenation; Structure, properties and uses of allotropes and oxides of carbon, silicon tetrachloride, silicates, zeolites and silicones.
Group - 15
                Properties and uses of nitrogen and phosphorus; Allotrophic forms of phosphorus; Preparation, properties, structure and uses of ammonia, nitric acid, phosphine and phosphorus halides,    (PCl3, PCl5); Structures of oxides and oxoacids of nitrogen and phosphorus.
Group - 16
              Preparation, properties, structures and uses of dioxygen and ozone; Allotropic forms of sulphur; Preparation, properties, structures and uses of sulphur dioxide, sulphuric acid (including its industrial preparation); Structures of oxoacids of sulphur.
Group - 17
                 Preparation, properties and uses of chlorine and hydrochloric acid; Trends in the acidic nature of hydrogen halides; Structures of Interhalogen compounds and oxides and oxoacids of halogens.
Group -18
                Occurrence and uses of noble gases; Structures of fluorides and oxides    of xenon.
UNIT 16: d – and f – BLOCK ELEMENTS
Transition Elements
General introduction, electronic configuration, occurrence and characteristics, general trends in properties of the first row transition elements - physical properties, ionization enthalpy, oxidation states, atomic radii, colour, catalytic behaviour, magnetic properties, complex formation, interstitial compounds, alloy formation; Preparation, properties and uses of K2Cr2O7 and KMnO4.
Inner Transition Elements
Lanthanoids - Electronic configuration, oxidation states, chemical reactivity and lanthanoid contraction.
Actinoids - Electronic configuration and oxidation states.
UNIT 17: CO-ORDINATION COMPOUNDS
Introduction to co-ordination compounds, Werner’s theory; ligands, co-ordination number, denticity, chelation; IUPAC nomenclature of mononuclear co-ordination compounds, isomerism; Bonding-Valence bond approach and basic ideas of Crystal field theory, colour and magnetic properties; Importance of co-ordination compounds (in qualitative analysis, extraction of metals and in biological systems).
UNIT 18: ENVIRONMENTAL CHEMISTRY
Environmental pollution - Atmospheric, water and soil.
Atmospheric pollution - Tropospheric and stratospheric
Tropospheric pollutants - Gaseous pollutants: Oxides of carbon, nitrogen and sulphur, hydrocarbons; their sources, harmful effects and prevention; Green house effect and Global warming; Acid rain;
Particulate pollutants: Smoke, dust, smog, fumes, mist; their sources, harmful effects and prevention.
Stratospheric pollution- Formation and breakdown of ozone, depletion of ozone layer - its mechanism and effects.
Water Pollution - Major pollutants such as, pathogens, organic wastes and chemical pollutants; their harmful effects and prevention.
Soil pollution - Major pollutants such as: Pesticides (insecticides,. herbicides and fungicides), their harmful effects and prevention.
Strategies to control environmental pollution.

 
Section-C

Organic Chemistry

UNIT 19: PURIFICATION AND CHARACTERISATION OF ORGANIC COMPOUNDS
Purification - Crystallization, sublimation, distillation, differential extraction and chromatography - principles and their applications.
Qualitative analysis - Detection of nitrogen, sulphur, phosphorus and halogens.
Quantitative analysis (basic principles only) - Estimation of carbon, hydrogen, nitrogen, halogens, sulphur, phosphorus.
Calculations of empirical formulae and molecular formulae; Numerical problems in organic quantitative analysis.

UNIT 20: SOME BASIC PRINCIPLES OF ORGANIC CHEMISTRY
Tetravalency of carbon; Shapes of simple molecules - hybridization (s and p); Classification of organic compounds based on functional groups: - C = C - , - C ? C - and those containing halogens, oxygen, nitrogen and sulphur; Homologous series; Isomerism - structural and stereoisomerism.
Nomenclature (Trivial and IUPAC)
Covalent bond fission - Homolytic and heterolytic: free radicals, carbocations and carbanions; stability of carbocations and free radicals, electrophiles and nucleophiles.
Electronic displacement in a covalent bond - Inductive effect, electromeric effect, resonance and hyperconjugation.
Common types of organic reactions - Substitution, addition, elimination and rearrangement.

UNIT 21: HYDROCARBONS
Classification, isomerism, IUPAC nomenclature, general methods of preparation, properties and reactions.
Alkanes - Conformations: Sawhorse and Newman projections (of ethane); Mechanism of halogenation of alkanes.
Alkenes - Geometrical isomerism; Mechanism of electrophilic addition: addition of hydrogen, halogens, water, hydrogen halides (Markownikoff’s and peroxide effect); Ozonolysis, oxidation, and polymerization.
Alkynes - Acidic character; Addition of hydrogen, halogens, water and hydrogen halides; Polymerization.
Aromatic hydrocarbons - Nomenclature, benzene - structure and aromaticity; Mechanism of electrophilic substitution: halogenation, nitration, Friedel – Craft’s alkylation and acylation, directive influence of functional group in mono-substituted benzene.
UNIT 22: ORGANIC COMPOUNDS CONTAINING HALOGENS
General methods of preparation, properties and reactions; Nature of C-X bond; Mechanisms of substitution reactions.
Uses; Environmental effects of chloroform, iodoform, freons and DDT.
UNIT 23: ORGANIC COMPOUNDS CONTAINING OXYGEN
General methods of preparation, properties, reactions and uses.
ALCOHOLS, PHENOLS AND ETHERS
Alcohols: Identification of primary, secondary and tertiary alcohols; mechanism of dehydration.
Phenols: Acidic nature, electrophilic substitution reactions: halogenation, nitration and sulphonation, Reimer - Tiemann reaction.
Ethers: Structure.
Aldehyde and Ketones: Nature of carbonyl group;
Nucleophilic addition to >C=O group, relative reactivities of aldehydes and ketones; Important reactions such as - Nucleophilic addition reactions (addition of HCN, NH3 and its derivatives), Grignard reagent; oxidation; reduction (Wolff Kishner and Clemmensen); acidity of ? - hydrogen, aldol condensation, Cannizzaro reaction, Haloform reaction; Chemical tests to distinguish between aldehydes and Ketones.
CARBOXYLIC ACIDS
Acidic strength and factors affecting it.

UNIT 24: ORGANIC COMPOUNDS CONTAINING NITROGEN
General methods of preparation, properties, reactions and uses.
Amines: Nomenclature, classification, structure, basic character and identification of primary, secondary and tertiary amines and their basic character.
Diazonium Salts: Importance in synthetic organic chemistry.
UNIT 25: POLYMERS
General introduction and classification of polymers, general methods of polymerization - addition and condensation, copolymerization; Natural and synthetic rubber and vulcanization; some important polymers with emphasis on their monomers and uses - polythene, nylon, polyester and bakelite.

UNIT 26: BIO MOLECULES
General introduction and importance of biomolecules.
CARBOHYDRATES - Classification: aldoses and ketoses; monosaccharides (glucose and fructose), constituent monosaccharides of oligosacchorides (sucrose, lactose, maltose) and polysaccharides (starch, cellulose, glycogen).
PROTEINS - Elementary Idea of ? - amino acids, peptide bond, polypeptides; Proteins: primary, secondary, tertiary and quaternary structure (qualitative idea only), denaturation of proteins, enzymes.
VITAMINS - Classification and functions.
NUCLEIC ACIDS - Chemical constitution of DNA and RNA.
Biological functions of nucleic acids.
UNIT 27: CHEMISTRY IN EVERYDAY LIFE
Chemicals in medicines - Analgesics, tranquilizers, antiseptics, disinfectants, antimicrobials, antifertility drugs, antibiotics, antacids, antihistamins - their meaning and common examples.
Chemicals in food - Preservatives, artificial sweetening agents - common examples.
Cleansing agents - Soaps and detergents, cleansing action.

UNIT 28: PRINCIPLES RELATED TO PRACTICAL CHEMISTRY
• Detection of extra elements (N,S, halogens) in organic compounds; Detection of the following functional groups: hydroxyl (alcoholic and phenolic), carbonyl (aldehyde and ketone), carboxyl and amino groups in organic compounds.
• Chemistry involved in the preparation of the following:
Inorganic compounds: Mohr’s salt, potash alum.
Organic compounds: Acetanilide, p-nitroacetanilide, aniline yellow, iodoform.
• Chemistry involved in the titrimetric excercises - Acids bases and the use of indicators, oxalic-acid vs KMnO4, Mohr’s salt vs KMnO4.
• Chemical principles involved in the qualitative salt analysis:
Cations - Pb2+ , Cu2+, AI3+, Fe3+, Zn2+, Ni2+, Ca2+, Ba2+, Mg2+, NH4+.
Anions- CO32-, S2-, SO42-, NO2-, NO3-, CI-, Br, I. (Insoluble salts excluded).
• Chemical principles involved in the following experiments:
1. Enthalpy of solution of CuSO4
2. Enthalpy of neutralization of strong acid and strong base. .
3. Preparation of lyophilic and lyophobic sols.
4. Kinetic study of reaction of iodide ion with hydrogen peroxide at room temperature.

SYLLABUS FOR APTITUDE TEST B.ARCH./B.PLANNING
 
Part I
Awareness of persons, places, Buildings, Materials.) Objects, Texture related to Architecture and build~environment. Visualising three dimensional objects from two dimensional drawings. Visualising. different sides of three dimensional objects. Analytical Reasoning Mental Ability (Visual, Numerical and Verbal).

Part II
Three dimensional - perception: Understanding and appreciation of scale and proportion of objects, building forms and elements, colour texture, harmony and contrast. Design and drawing of geometrical or abstract shapes and patterns in pencil. Transformation of forms both 2 D and 3 D union, substraction, rotation, development of surfaces and volumes, Generation of Plan, elevations and 3 D views of objects. Creating two dimensional and three dimensional compositions using given shapes and forms. Sketching of scenes and activities from memory of urbanscape (public space, market, festivals, street scenes, monuments, recreational spaces etc.), landscape (river fronts, jungles. gardens, tre es, plants etc.) and rural life.

Note:
Candidates are advised to bring pencils, own geometry box set, erasers and colour pencils and crayons for the Aptitude Test.
For More Help, Visit here
Contact:
The JOINT SECRETARY (AIEEE UNIT)
CENTRAL BOARD OF SECONDARY EDUCATION
SHIKSHA SADAN, 17, ROUSE AVENUE,INSTITUTIONAL AREA,
NEAR BAL BHAWAN, NEW DELHI - 110002.
TELEPHONE : 011-23220156, FAX : 011-23220152
[Courtsey: AIEEE Offical Site]

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Online Math and Science Tutors from India

Using computer whiteboards and talking through the computer, many U.S. students are now finding that they can get knowledgeable math and science tutors from India.

Excerpt:"Both Growing Stars and Studyloft say that the majority of their tutors hold master's degrees or doctorates in their subjects; many also have degrees in education."

The idea is that tutors through the Internet relieve parents of commuting and scheduling, and cut the prices of established tutoring businesses like Sylvan Learning. If any of you have experiences with an online tutoring service(good or bad), please leave us a comment. Thanks in advance!

Online tutoring - The Boston Globe
Studyloft.com
Growingstars.com

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Decimal Point Pickle

I came up with a variation on what was already a decent game and got to pilot it with Mr. Schiller's 5th grade this week.  Esther Billings introduced me to the game she had found in the book, Nimble with Numbers by Leigh Childs and Laura Choate, Dale Seymour Pub., 1998

The 5th grade came up with two names, Destination Elimination (which I like because it rhymes), and Decimal Pickle.  This suggested by a student who's answer for everything is pickle.  (I'm sure you know a student like that.)  But here, it reminded me of a childhood baseball game that none of the kids knew but kind of fits.  (The baseball game Pickle.)

 My Favorite Pickle

Decimal Point Pickle

Set Up:
1.    2 or more teams or players.
2.    Get a deck of cards and remove the Kings, Queens, 10s and Jokers.  Jacks stay in.
3.    Each player or team makes a path with 10 spaces.  It can be straight and rectangles, or it can be curvy and circles, but it needs to have 10 spaces and a clear beginning and end.
4.    Shuffle the cards.

Playing:  Idea is that you’re going to fill in your path from small to big, flipping over cards to get possibilities.
1.    On your turn, flip over a card.  If it’s red, flip over another card.  If it’s red, flip over another card.  But you never flip more than three.  If you run out of cards, shuffle up the used cards.
2.    Arrange those cards to make a decimal number.  Jacks are the zeros. The smallest number you can make is .000, and the largest is .999.  Say your number.
3.    Fill in your decimal number somewhere on the path.  But it can’t go before a smaller number or after a bigger number.  Your path has to start small and end big.  If there’s no place to fill in your number, you don’t. 
4.    Winner is the first person to completely fill in their path, with all the numbers in order.

Examples:
1.    J ♥, 3 ♣.  You can make .03 or .30. 
2.    5 ♥ hearts, so you flip 2 ♦, so you flip 7 ♥ hearts.  (You stop because you can’t have more than three.)  You can make one of .275, .275, .527, .572, .725 or .752.  Which you want depends on your path. 
3.    Sample filled in path below.

Variations: 
1.    Simpler:  Play where you always flip over 2 or 3 cards.
2.    Play cooperatively.  Two players work together to fill in one path.
3.    More complex:  Play with 10s, which fill in 2 places.  So 10 ♦, 5 ♠ can be .105 or .510.
4.    More complex:  Play without the three card limit.  You could hit a 10 digit long decimal or longer!  (Pretty unlikely, but still…)
5.    Make 12 space paths.
6.    Play with Jokers as a wild digit.

Teaching Notes:  As often with a new game I played me vs the class first.  It was clear that the blackjack-esque possibility of extra cards was exciting, and they quickly got the idea that it was a big advantage.  I didn't castigate anyone for saying "point two three" but often asked "so how do you say that number?"  I shared how I thought about getting numbers close together and they really ran with it.  In general hitting on lots of ideas about where to put numbers, how to divide up the path, etc.  In their 2 on 2 games, there was a lot of good discussion about strategy, how to leave space, and what they wanted to turn over.  There was a lot of excellent comparing of decimals of different length.  (One amazing discussion comparing .1 to .065)  Students got very creative with their paths and I was quite glad I hadn't brought any preprinted ones.  We actually wound up playing with everyday math cards, which thankfully ! came in black and blue.  Whew!

If you give it a try, please let me know what you think.


<--- 2nd favorite pickles

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Free Math Help.com

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Algebra 2 for Dummies for Homeschool

 Of all of the Algebra 2 curriculum options available!  My son started the year with an Algebra 2 workbook and Khan Academy for Algebra 2.  

After a month or two, he came to me and asked for Algebra 2 for Dummies.  "For Dummies!?"  I asked.... "For Dummies" he replied.  I purchased the set... a textbook and a workbook.

He completed it today and we went over his work together.  Here is how he used it.

First, he read the chapter.  Then if he wasn't clear on it, he Googled the concept and looked at videos and sites like AAAmath.com.  Then he printed a free worsheet on the topic.  Finally, when he felt comfortable, he used the workbook for his chapter tests.

His reasoning for his approach is that Algebra 2 for Dummies should be used as a review for someone who has already taken the class, or for extra help for someone taking the class.  Not as a first exposure. Still, that and the internet was all he needed

He scored a 91% for the year... starting off in the high 80's and then got better as he progressed.  He tackled this subject with NO help from me... I even offered to hire a tutor many times when he announced he was stuck.  He politely declined and soldiered on.  I think he is as proud of his accomplishment as I am of him.

Now on to Trig!

He is taking it as a quick summer course using online videos and this book:




~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Visit My Education Column at Examiner.com Check out FREE HOME EDUCATION WEBSITE for free homeschool links and resources

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CBSE 2010 Class XI (2008-09) Syllabus

PHYSICS (Code No. 042)

Senior Secondary stage of school education is a stage of transition from general
education to discipline-based focus on curriculum. The present updated syllabus keeps
in view the rigour and depth of disciplinary approach as well as the comprehension
level of learners. Due care has also been taken that the syllabus is not heavy and is at
the same time, comparable to the international standards. Salient features of the syllabus
include:
_ Emphasis on basic conceptual understanding of the content.
_ Emphasis on use of SI units, symbols, nomenclature of physical quantities and
formulations as per international standards.
_ Providing logical sequencing of units of the subject matter and proper placement of
concepts with their linkage for better learning.
_ Reducing the curriculum load by eliminating overlapping of concepts/ content within
the discipline and other disciplines.
_ Prom! otion of process-skills, problem-solving abilities and applications of Physics
concepts.

Besides, the syllabus also attempts to
_ strengthen the concepts developed at the secondary stage to provide firm foundation
for further learning in the subject.
_ expose the learners to different processes used in Physics-related industrial and
technological applications.
_ develop process-skills and experimental, observational, manipulative, decision
making and investigatory skills in the learners.
_ promote problem solving abilities and creative thinking in learners.
_ develop conceptual competence in the learners and make them realize and appreciate
the interface of Physics with other disciplines.


COURSE STRUCTURE

Class XI (Theory)
One Paper Three Hours Max Marks: 70
Class XI Weightage
Unit I Physical World & Measurement 03
Unit II Kinematics 10
Unit III Laws of Motion 1! 0
Unit IV Work, Energy & Power 06
Unit V Motion of S! ystem of particles & Rigid Body 06
Unit VI Gravitation 05
Unit VII Properties of Bulk Matter 10
Unit VIII Thermodynamics 05
Unit XI Behaviour of Perfect Gas & Kinetic Theory of gases 05
Unit X Oscillations & Waves 10
Total 70

Unit I: Physical World and Measurement (periods 10)

Physics - scope and excitement; nature of physical laws; Physics, technology and society.
Need for measurement: Units of measurement; systems of units; SI units, fundamental
and derived units. Length, mass and time measurements; accuracy and precision of
measuring instruments; errors in measurement; significant figures.
Dimensions of physical quantities, dimensional analysis and its applications.

Unit II: Kinematics (Periods 30)

Frame of reference. Motion in a straight line: Position-time graph, speed and velocity.
Uniform and non-uniform motion, average speed and instantaneous velocity.
Uniformly accelerate! d motion, velocity-time, position-time graphs, relations for uniformly accelerated motion (graphical treatment).
Elementary concepts of differentiation and integration for describing motion.
Scalar and vector quantities: Position and displacement vectors, general vectors and
notation, equality of vectors, multiplication of vectors by a real number; addition and
subtraction of vectors. Relative velocity.
Unit vector; Resolution of a vector in a plane - rectangular components. Motion in a
plane. Cases of uniform velocity and uniform acceleration-projectile motion. Uniform
circular motion.

Unit III: Laws of Motion (Periods 16)

Intuitive concept of force. Inertia, Newton’s first law of motion; momentum and Newton’s second law of motion; impulse; Newton’s third law of motion. Law of conservation of linear momentum and its applications.
Equilibrium of concurrent forces. Static and kinetic friction, la! ws of friction, rolling friction.
Dynamics of uniform cir! cular mo tion: Centripetal force, examples of circular motion
(vehicle on level circular road, vehicle on banked road).

Unit IV: Work, Energy and Power (Periods 16)

Scalar product of vectors. Work done by a constant force and a variable force; kinetic
energy, work-energy theorem, power.
Notion of potential energy, potential energy of a spring, conservative forces: conservation
of mechanical energy (kinetic and potential energies); non-conservative forces: elastic
and inelastic collisions in one and two dimensions.

Unit V: Motion of System of Particles and Rigid Body (Periods 18)

Centre of mass of a two-particle system, momentum conversation and centre of mass
motion. Centre of mass of a rigid body; centre of mass of uniform rod.
Vector product of vectors; moment of a force, torque, angular momentum, conservation
of angular momentum with some examples.
Equilibrium of rigid bodies, rigid body rotat! ion and equations of rotational motion,
comparison of linear and rotational motions; moment of inertia, radius of gyration.
Values of moments of inertia for simple geometrical objects (no derivation). Statement of
parallel and perpendicular axes theorems and their applications.

Unit VI: Gravitation (Periods 14)

Keplar’s laws of planetary motion. The universal law of gravitation.
Acceleration due to gravity and its variation with altitude and depth.
Gravitational potential energy; gravitational potential. Escape velocity. Orbital velocity
of a satellite. Geo-stationary satellites.

Unit VII: Properties of Bulk Matter (Periods 28)

Elastic behaviour, Stress-strain relationship, Hooke’s law, Young’s modulus, bulk modulus, shear, modulus of rigidity.
Pressure due to a fluid column; Pascal’s law and its applications (hydraulic lift and hydraulic brakes). Effect of gravity on flui! d pressure.
Viscosity, Stokes’ law, terminal veloci! ty, Reyn old’s number, streamline and turbulent
flow. Bernoulli’s theorem and its applications.
Surface energy and surface tension, angle of contact, application of surface tension ideas to drops, bubbles and capillary rise.
Heat, temperature, thermal expansion; specific heat - calorimetry; change of state - latent heat.
Heat transfer-conduction, convection and radiation, thermal conductivity, Newton’s law of cooling.

Unit VIII: Thermodynamics (Periods 12)

Thermal equilibrium and definition of temperature (zeroth law of thermodynamics). Heat, work and internal energy. First law of thermodynamics.
Second law of thermodynamics: reversible and irreversible processes. Heat engines and
refrigerators.

Unit IX: Behaviour of Perfect Gas and Kinetic Theory (Periods 8)

Equation of state of a perfect gas, work done on compressing a gas.
Kinetic theory of gases - assumptions, concept of pressure. Kin! etic energy and temperature; rms speed of gas molecules; degrees of freedom, law of equipartition of energy (statement only) and application to specific heats of gases; concept of mean free path, Avogadro’s number.

Unit X: Oscillations and Waves (Periods 28)

Periodic motion - period, frequency, displacement as a function of time. Periodic functions.
Simple harmonic motion (S.H.M) and its equation; phase; oscillations of a spring–restoring force and force constant; energy in S.H.M.-kinetic and potential energies; simple pendulum–derivation of expression for its time period; free, forced and damped oscillations (qualitative ideas only), resonance.
Wave motion. Longitudinal and transverse waves, speed of wave motion. Displacement
relation for a progressive wave. Principle of superposition of waves, reflection of waves,
standing waves in strings and organ pipes, fundamental mode and harmonics, Beats,
Doppler effec! t.

Practicals
Note: Every student will perform! 10 expe riments (5 from each section) and 8 activities (4
from each section) during the academic year.
Two demonstration experiments must be performed by the teacher with participation of
students. The students will maintain a record of these demonstration experiments. Schools are advised to see the guidelines for evaluation in practicals for Class XII. Similar pattern may the followed for Class XI.

SECTION A
Experiments
1. Use of Vernier Callipers
(i) to measure diameter of a small spherical/cylindrical body.
(ii) to measure dimensions of a given regular body of known mass and hence find its
density.
(iii) to measure internal diameter and depth of a given beaker/calorimeter and hence
find its volume.
2. Use of screw gauge
(i) to measure diameter of a given wire, (ii) to measure thickness of a given sheet
(iii) to measure volume of an irregular lamina
3. To determine radius of curvature of a given spher! ical surface by a spherometer.
4. To find the weight of a given body using parallelogram law of vectors.
5. Using a simple pendulum, plot L-T and L-T2 graphs. Hence find the effective length of
second’s pendulum using appropriate graph.
6. To study the relationship between force of limiting friction and normal reaction and to find co-efficient of friction between a block and a horizontal surface.
7. To find the downward force, along an inclined plane, acting on a roller due to gravitational pull of the earth and study its relationship with the angle of inclination by plotting graph between force and sin.

Activities

1. To make a paper scale of given least count, e.g. 0.2cm, 0.5cm.
2. To determine mass of a given body using a metre scale by principle of moments.
3. To plot a graph for a given set of data, with proper choice of scales and error bars.
4. To measure the force of limiting friction for rol! ling of a roller on a horizontal plane.
5. To study the v! ariation in range of a jet of water with angle of projection.
6. To study the conservation of energy of a ball rolling down on inclined plane (using a
double inclined plane).
7. To study dissipation of energy of a simple pendulum by plotting a graph between square of amplitude and time.

SECTION B

Experiments

1. To determine Young’s modulus of elasticity of the material of a given wire.
2. To find the force constant of a helical spring by plotting graph between load and extension.
3. To study the variation in volume with pressure for a sample of air at constant temperature by plotting graphs between P and V, and between P and I/V.
4. To determine the surface tension of water by capillary rise method.
5. To determine the coefficient of viscosity of a given viscous liquid by measuring terminal velocity of a given spherical body.
6. To study the relationship between the temperature of a hot body and time by p! lotting a
cooling curve.
7. (i) To study the relation between frequency and length of a given wire under constant
tension using sonometer.
(ii) To study the relation between the length of a given wire and tension for constant
frequency using sonometer.
8. To find the speed of sound in air at room temperature using a resonance tube by two resonance positions.
9. To determine specific heat of a given (i) solid (ii) liquid, by method of mixtures.

Activities

1. To observe change of state and plot a cooling curve for molten wax.
2. To observe and explain the effect of heating on a bi-metallic strip.
3. To note the change in level of liquid in a container on heating and interpret the observations.
4. To study the effect of detergent on surface tension by observing capillary rise.
5. To study the factors affecting the rate of loss of heat of a liquid.
6. To study the effect of load on depression of a! suitably clamped metre scale loaded
(i) at its end (ii) ! in the m iddle.

Recommended Textbooks.

1. Physics Part-I, Textbook for Class XI, Published by NCERT
2 . Physics Part-II, Textbook for Class XI, Published by NCERT

force and laws of motion class 9

GMAT Number Properties

Adapted from various sources of BeatTheGMAT

#1. How to test whether a number is prime or composite:

A Prime Number is a positive integer that is divisible by ONLY 2 numbers (1 and itself).
Whereas, A composite number is a positive integer which has divisor(s) other than the 2 numbers (1 and itself).

Following are the steps to test whether a number is a prime or composite,

1. Identify the perfect square (P.S) closest to the n < n.
2. Compute the square root of P.S
3. List all prime numbers upto the computed square root
4. Check if all listed prime numbers divide n equally. If not, then n is a prime. Even if atleast one of the listed prime numbers divide n, then n is a composite.

Example:

Take n as 113. To test whether 113 is a prime,

1. 100 is the closest perfect square to 113 (Remember that you take a closest perfect square that is smaller than n itself!)
2. Square root of! 100 ==> 10
3. Prime numbers upto the square root (10) ==> 2,3,5,7.
4. Check whether 2,3,5,7 divides 113. None of the numbers divide 113. So, 113 is a prime.

#2. How to calculate LCM and HCF of fractions:
L.C.M of 2 fractions = L.C.M of NUMERATORS / H.C.F of DENOMINATORS
H.C.F of 2 fractions = H.C.F of NUMERATORS / L.C.M of DENOMINATORS
Product of any 2 numbers = Product of LCM and HCF of those 2 numbers
Product of any 2 fractions = Product of LCM and HCF of those 2 fractions

Highest common factor of two or more than two numbers is the greatest number that divides each of them exactly (without remainder).

Ex. Find HCF of 18, 20 and 36
18 = 2 × 3^2
20 = 2^2 × 5
36 = 2^2 × 3^2 Therefore HCF = 2

The smallest number divisible by all the numbers is the Least Common Multiple of the given numbers.

Ex What is the LCM of 4, 8 and 12?
4 = 2^2, 8 = 2^3, 12 = 2^2 x 3!
Therefore LCM = 2^3 x 3 = 24

Q1: Find the L.! C.M. of 36, 48, 72 and 108. = 432
Q2: Find the greatest number which on dividing 277, 757 and 3307 leaves the same remainder 7 in each case. (Hint: Find HCF of numbers minus 7 = 30)
Q3: The L.C.M. of two number is 252 and their ratio is 6:7. Find the numbers. = 36,42
Q4: Find the greatest number that will divide 63, 45 and 69 so as to leave the same remainder (options 6,9,8,10). Hint find HCF = 6
Q5: A room is 2.47m long and 2.09m broad. It is required to pave the floor with minimum square slabs. Find the number of slabs required for floor. - 143
Q6: Find the least perfect square number which is divisible by 2,3,4,5,6. - 900
Q7: Find the least five digit number which on divided by 12, 18, 21, and 28 leaves the same remainder - 11019, 12089, 10119, 10059. - 10119
Q8: Which smallest number must be added to 605329 so that it becomes divisible by 9? - 2
Q9; Find the greatest number of four digits which is exactly divisible by each of 12, 18, 40 an! d 45. - 9720
Q10: Find the L.C.M. of 6/7, 12/13, 7/15 - 84
Q11: Find the least number which is exactly divisible by 18, 15, 25 and 30. - 450
Q12: The sum and the differences of the L.C.M. and the H.C.F. of two numbers are 312 and 264, respectively. Find the numbers if their sum is 168. - 96,72

Source: http://www.quizmoz.com/quizzes/GMAT-Practice-Tests/g/GMAT-Practice-Test-Math-LCM-and-HCF.asp


Solutions:
A1. 36=2^2.3^2, 48=2^4.3^1, 72=2^3.3^2, 108=2^2.3^3
LCM = 2^4.3^3 = 27*2*2*2*2 = 54*2*2*2 = 108*2*2 = 216*2 = 432

A2.
277-7 = 270 = 2^1.3^3.5^1
757-7 = 750 = 2^1.3^1.5^3
3307-7 = 3300 = 2^2.3^1.5^2.11^1
HCF = 2^1.3^1.5^1 = 30

A3.
252 = 2^2.3^2.7 = 6*6*7
Thus, use 6*6 and 6*7 = 36, 42

A4: Try choices, remainders are
(1) 6: 3, 3, 3
(2) 9: 0, 0, 6
(3) 8: 7, 5, 5
(4) 10: 3, 5, 9

A5: Area = 2! .47*2.09 m^2 = 247*209 cm^2 = n*x^2 where x is the HCF in 247 ! and 209< br />Q: Is 247 or 209 prime
Prime #s - 2,3,5,7,11,13,17,19,23,27
247: Closest square = 225 = 15^2 => Check 2,3,5,7,11,13 => 13 divides 247 => 247 is a composite number = 13*19
209: Closest square = 196 = 14^2 => Check 2,3,5,7,11,13 => 11 divides 209 => 209 is also a composite number = 11*19
Thus, Area = 11*13*19^2 => x=19 cm and n = 11*13
Therefore, # of slabs = 11*13 = 143 with each having 19 cm side

A6: Want to find least perfect square number which is divisible by 2,3,4,5,6
Number have the same prime factor two times
2 = 2^1.3^0.5^0
3 = 2^0.3^1.5^0
4 = 2^2.3^0.5^0
5 = 2^0.3^0.5^1
6 = 2^1.3^1.5^0

2^2.3^2.5^2 = 4.9.25 = 900

#3. How to find unit digit of powers of numbers:
All numbers from 0 to 9 have a repetitive pattern for powers

Pattern 1:
Unit's place that has digits - 2/3/7/8
Then, unit's digit repeats every 4th value.! Divide the power (or index) by 4.

After dividing,
If remainder is 1, unit digit of number raised to the power 1.
If remainder is 2, unit digit of number raised to the power 2.
If remainder is 3, unit digit of number raised to the power 3.
If remainder is 0, unit digit of number raised to the power 4.

Pattern 2:
Unit's place that has digits - 0/1/5/6
Then, all powers of the number have same digit as unit's place.
For e.g., 6^1 = 6, 6^2 = 36, 6^3 = 216, 6^4 = 1296


Pattern 3:
Unit's place that has digit - 4
Then,
If power is odd --> unit's digit will be '4'
If power is even --> unit's digit will be '6'

Similarly,
Unit's place that has digit - 9
Then,
If power is odd --> unit's digit will be '9
If power is even --> unit's digit will be '1'!

Ex. OG-12 PS190
190. What is the units digit ! of (13)^ 4(17)^2(29)^3 ?
powers of 3
3
9
7
1
7
9

9
1
9

1*9*9 = 1

#4. Divisibility Tests:

To check whether a number (say n) is divisible

By 2: unit's place of n must be 0 (OR) unit's place of n must be divisible by 2.
By 3: Sum of the digits of n must be divisible by 3.
By 4: Last 2 digits (Unit's place and ten's place) of n are 0's (OR) Last 2 digits of n must be divisible by 4.
By 5: Unit's digit must be a 5 (OR) a 0.
By 6: n must be divisible by both 2 and 3 (Follow the method used for 2 and 3).
By 8: Last 3 digits (units, tens and hundredth place) of n are 0's (OR) Last 3 digits of n is divisible by 8.
By 9: Sum of the digits of n must be divisible by 9.
By 11: (Sum of the digits of n in odd places) - (Sum of the digits of n in even places) ==> Either 0 (OR) divisible by 11.
By 12: n must be divisible by ! both 3 and 4 (Follow the method used for 3 and 4).
By 25: Last 2 digits (units and tens place) of n are 0's (OR) Last 2 digits of n must be divisible by 25.
By 75: n must be divisible by both 3 and 25 (Follow the method used for 3 and 25).
By 125: Last 3 digits of n are 0's (OR) are divisible by 125.

#5. Whether a number is NOT a perfect square or MIGHT BE

The perfect squares always end in 1, 4, 5, 6, 9 or 00 (i.e. Even number of zeros).

If they end in even zeros, then remove the zeros at the end of the number and apply following tests:
If unit digit ends in 5, ten’s digit is always 2.
If unit digit ends in 6, ten’s digit is always odd
otherwise it is always even.

Digital roots are 1, 4, 7 and 9.
To find digital root of a number, add all its digits.
If this sum is more than 9, add the digits of this sum. The single digit obtained at the! end is the digital root of the number.

If a number! is divi sible by 4, its square leaves a remainder 0 when divided by 8. Square of an even number not divisible by 4 leaves remainder 4 while
square of an odd number always leaves remainder 1 when divided by 8.

Total numbers of prime factors of a perfect square are always odd.

Examples:
1. 5329 = sum of digits = 19 = 1 => Digital root is in (1,4,7,9) => MAYBE
2. 1280 = NO because does not end in 1,4,5,6,9,00
3. 1800 = Ends in 00, sum of digits = 9, unit digit not in (5,6), tens
4. 3600 = Ends in 00, sum of digits = 9, unit digit = 6, tens digit = odd => MAYBE
5. 729 = sum of digits = 18 = 9 => Digital root is in (1,4,7,9)
6. 38910 = No because does not end in 1,4,5,6,9,00
7. 14798678562 = No because does not end in 1,4,5,6,9,00
8. 15763530163289 = sum of digits = 58 = 13 = 4 digital root ends in (1,4,7,9) MIGHT BE
Ex. 1800 = 9 YES =>

dividing three digit single digit number

Learning a Lesson with a Looking Glass

Happy Ascension! Today might be said to be the feast day of mathematics, since it was the day Jesus told His apostles to go out and matheteusate all nations... The Greek was translated to the Latin discipuli which usually is left as "disciples" - but it has another English translation: "the Learners".

Yes, as I mentioned recently, math is easy - and today being its feast day, I thought we'd look at another odd but common truth which you would never guess is mathematical, yet is one of its most important stones in its foundation: the truth we call Right and Left...

((click here to proceed))

Of course you want to hear some Chesterton before you suffer through such an odd affair, so I will supply it:

[Syme the policeman poet said] "...So it is in mere curiosit! y that I make my queries. First of all, what is it really all about? What is it you object to? You want to abolish government?"
"To abolish God!" said Gregory [the anarchist] opening the eyes of a fanatic. "We do not only want to upset a few despotisms and police regulations; that sort of anarchism does exist, but it is a mere branch of the Nonconformists. We dig deeper and we blow you higher. We wish to deny all those arbitrary distinctions of vice and virtue, honour and treachery, upon which mere rebels base themselves. The silly sentimentalists of the French Revolution talked of the Rights of Man! We hate Rights as we hate Wrongs. We have abolished Right and Wrong."
"And Right and Left," said Syme with a simple eagerness, "I hope you will abolish them too. They are much more troublesome to me."
[GKC The Man Who Was Thursday CW6:490]

In order to set my stage, I shall invoke another text, sometimes very Chestertonian in form, and one of my! favourite sources of delight: the famous "Calvin and Hobbes" ! comics o f Bill Watterson. If you have the collection called Attack of the Deranged Mutant Killer Monster Snow Goons you will find them on pages 35 and 37. Yes, perhaps they are on-line somewhere, but I cannot take the time to find them, so I will merely give you the dialog. You must bear in mind that "Calvin" is a six-year-old boy and "Hobbes" is his stuffed tiger who often speaks with Calvin when they are alone...

Calvin: (to Hobbes) Help me with this homework, OK? What 6 + 3?
Hobbes: 6+3, eh? Well, this one is a bit tricky. First we call the answer "Y" as in "Y do we care?" Now Y may be a square number, so we'll draw a square and make this side 6 and that side 3. Then we'll measure the diagonal. (He draws the square and labels it.)
Calvin: (staring at the drawing) I don't remember the teacher explaining it like this.
Hobbes. (waving the pencil knowingly) She probably doesn't know higher math. When you deal with high numbers, you need higher m! ath.
Calvin: (measures the diagonal) But this diagonal is just a little under two.
Hobbes: OK. Here, I'll draw a bigger square.

Now, assuming you've stopped laughing, you are wondering what this has to do with Chesterton. Oh, ye of little faith! How soon you have forgotten this important line:

Men can construct a science with very few instruments, or with very plain instruments; but no one on earth could construct a science with unreliable instruments. A man might work out the whole of mathematics with a handful of pebbles, but not with a handful of clay which was always falling apart into new fragments, and falling together into new combinations. A man might measure heaven and earth with a reed, but not with a growing reed.
[GKC Heretics CW1:117]

Obviously, Hobbes is proposing to do just that. In the intervening pages, Calvin's mom goes to meet with his teacher, producing some very funny allusions much in the s! tyle of A. C. Doyle's references in the "Sherlock Holmes" stor! ies to m ysterious cases about which we never hear anything more... (All C&H readers recall the "Noodle Incident" and all those sirens at noon...) Be that as it may, Calvin's father takes a hand and tries to instruct his son in the far greater mystery of simple addition:

Dad: Here, maybe this will make more sense. (pun in the text) I have eight pennies. I ask you for four more.
Calvin: I say forget it. You're the one with a steady paycheck.
Dad: Just give me four pennies. Good. How much money do I have now?
Calvin: Investments and all?
Dad: (exasperatedly) No. Just here on the table.
Calvin: Eight cents.
Dad: No, eight plus four is twelve, see? Count them up.
Calvin: But those four are mine!

Ah... yes, please finish laughing. And now attend.

What was Calvin's mistake? What was his father's mistake? What is going on here? Are we talking about money - investments and all - or something else? Is this the b! eginning of a sermon on ownership and property, a commentary on the Commandments, or an elaboration of Leo XIII's Rerum Novarum with reference to Distributivism, Distributism, or something similar?

No. (Though it could be... but not this time.) It is my attempt at trying to awaken you to a very mysterious property, one of the most well-known relations of all the relations of things that we know of: the property of adjacency, of "things-being-next-to-each-other" - that is, of the mystery of Right and Left. Calvin's father is a lawyer and so he never studied the special branch of mathematics called "Set Theory" - which can go by other names like "Discrete Math" or "Number Theory". I am not going to scare you, but I want you to read on, and try very hard to start noticing something you are seeing constantly - and then realize that there was once a time that you did not yet grasp it, as Calvin did not grasp it... and finally you learned it. Alas! , you learned it too well, for it became a commonplace, and yo! u forgot about it... until a lunatic like me comes along and reminds you that it is still there.

One of the things we are taught - I mean we computer people, and we scientists, and we mathematicians - is to pay attention to detail, and try to see things as they are. That is very Chestertonian, and you no doubt have heard me quote such things before:

I would insist that people should have so much simplicity as
would enable them to see things suddenly and to see things as they are.
[GKC ILN July 13 1907 CW27:506]
...the object of my school is to show how many extraordinary things even a lazy and ordinary man may see if he can spur himself to the single activity of seeing.
[GKC Tremendous Trifles 6]

Yes. Now here is the idea I want you to see: What do we mean by addition? Can there be various kinds of such a basic and obvious thing?

It is so dreadfully obvious to us in computing, since we have to work with! a MACHINE, which does NOT understand, and will not EVER understand... yes, it may be hard for you to realize, but we can do NOTHING AT ALL with computers unless we use the very small collection of tools which are part of the machine - these are symbols, actually, which stand for certain simple kinds of actions or tasks. Most people walk around assuming computers can do "math" - by which they mean adding and stuff like that.

Here I must be Scholastic, and shout, with a blow on the table, distinguo! (I distinguish!)

Of course they do NOT add. No more than a smaller river "adds" when it pours its water into a larger one at the confluence of the waters. (I thank my good friend lUkE for this analogy; you may also recall a Tom Petty song about the river kissing the ocean...)

But as interesting (or dull) as this question is, we are not going to talk about that today and here. It would take a while, too, and though it is interesting, we n! eed to get at something far more interesting and hard to expla! in - the idea that Calvin's father was trying to teach.

What do we mean by addition? What do we mean - I use this word "mean" here and I do not know what other word to use - what do we mean when we place something next to something else and try to consider them in the joint sense, as something which is now single, though paradoxically it remains composite and multiple?

Is this sounding fantastic? Or confusing? Oh, why am I struggling to write about this as I wish?

Pardon me while I invoke my muse...

Oh dear little Alice, whose name is Truth:
Oh thou who dwellest in the Wonderland of the Real World,
Call upon the Holy Spirit, Who pours out fire in sevenfold stream on those who beseech His aid,
and give me the words that will bring true light. Amen.

Ah... I could try all day to explain this, and yet fail - even if I tried with both hands...

AHA.

"I'm sure I didn't ! mean" Alice was beginning, but the Red Queen interrupted her impatiently.
"That's just what I complain of! You should have meant! What do you suppose is the use of a child without any meaning? Even a joke should have some meaning - and a child's more important than a joke, I hope. You couldn't deny that, even if you tried with both hands."
"I don't deny things with my hands," Alice objected.
"Nobody said you did," said the Red Queen. "I said you couldn't if you tried."
[Lewis Carroll, Through the Looking Glass, chapter 9 "Queen Alice"]

Ah. Now we have something. Let us try with both hands.

(I must control myself here - the hand is a wonderful thing, and there are books about it, not counting those on anatomy, but I must not write another just now.)

In order to proceed, let us talk about another kind of number: numbe! r as used the grammarians, which means whether a word is "! singular " (that is, just one) or "plural" (that is, more than one). This idea is so dramatic and so penetrating of thought that many languages embed the "number" in a word as an ending or suffix. (Some, like Vietnamese, use another method.)

But did you know that Greek and Egyptian and Hebrew have a third form of this grammatical number? Yes - besides "one" or "many" they had a special word for "two" of a thing. Even Latin and English have traces of this powerful urge to call attention to those very special things that come in pairs, like hands and gloves, feet and shoes - twins of various kinds.

There is something very interesting about our hands, and I might talk about the important idea of "chirality" (which comes from the Greek word for "hand") in chemistry, and note how most of the molecules of living things are "handed" - the amino acids are all "left-handed", and there are sugars named from the Latin words for right (dextrose) and left (levulose! ) - and then there are things in physics like the "right-hand rule" for electric current and magnetism, or the "spins" of subatomic particles... what a world. No wonder Gabriel Syme found them troublesome!

But there is another aspect of our hands which provides the foundation for addition, the one which Calvin's father was trying to teach his son... From an incredibly young age, we get used to the idea that we have "sides" to our bodies, and that we can place things on our left or on our right, and that we bring things together by reaching out our hands and then pulling inwards - we carry or bring back those two separate things, and by our own hands we make them ONE.

Hence, by our "trying with both hands" we take two things and "bring them back" to be one... In Latin, they said relatio. That is what a relation is, the bringing back, of two things, to be one thing.

Now, immediately there springs up a quirk, which is what neithe! r Calvin nor his father grasped, and why I have needed to writ! e about this at such length. Dad had eight pennies, Calvin had four. The pennies are put together, but in what kind of "together" were they put? Calvin saw ownership persisting even in the collection - perhaps those were his four rare Indian-head pennies, and did not want to lose sight of them. Calvin's dad saw the dreary summation he kept a calculator to help him with, and merely counted the collection as "twelve".

There are other views. If you look at digits rather than words, and try adding - no let us say placing together - Dad's 8 with Calvin's 4, you might say the result is 84 or maybe 48. Here you show a mystic wisdom, for this is "addition" of another form, know to computer scientists as "concatenation" or more formally "addition of the free monoid over a finite alphabet"... but I am scaring you with these terms.

Do not be scared. This is a good insight, and points out Syme's struggle with right and left. You most likely know that Hebrew and Arabic a! re written from right to left. Perhaps you have heard that there was a form of ancient Greek that was written in "boustrophedon" = as the cow plows, or as we might say these days, as the ink-jet printer prints: on the first line print from left-to-right, and then go down and print from right-to-left. But, you scream, there is a big difference between "dog" and "god"! Correct. And from the gloomy halls that you just ran from, the theoretician responds:
"That is because the free monoid is not commutative."

You wanted an answer, now try with both hands. But Chesterton also has answered for us:

I remember a man ... who told me he was on a spiritual plane ("we are on different planes") on which yes and no, black and white, right and wrong, right and left, were all equal. I regarded him as I should any boastful aviator who told me that from the height to which he had risen all London looked like an exact chess-board, with all the squares and streets th! e same size. In short, I regarded him as a liar. London street! s are no t equally long, seen from a flying-ship or from anywhere else. And human sins or sorrows are not equally serious, seen in a vision or anywhere else.
[GKC ILN Aug 15 1914 CW30:145]

Yes, Alice, there is a difference between left and right. The amino acids in living things are "left-handed". If someone brewed a batch of right-handed amino acids in a lab, they would be chemically identical, and contain nothing extra, and have nothing left out - yet no earthly thing could use them to build proteins, any more than a right shoe can go onto a left foot.

And that is the great mystery I am trying to reveal. It is not only the very living parts of our bodies which possess handedness, but also the powerful and splendid human invention of letters and numbers.

We need to see this, as hard as it may be. And if you need a further demonstration,

!rorrim a tuohtiw siht daer ot (sdnah htob htiw) yrT

Even the very letters and number! s use so casually have their right and left hands - a separation as final as the Last Judgement:

And he shall set the sheep on his right hand, but
the goats on his left. [Matt 25:33]

Therefore what God has separated, let no calculator join...

discrete math symbols